This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TIE.2016.2531022 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS
1
Continuous-Discrete Time Observer Design for State and Disturbance Estimation of Electro-Hydraulic Actuator Systems Sofiane Ahmed Ali, Arnaud Christen, Steven Begg and Nicolas Langlois
Abstract—In this paper, a continuous-discrete time observer which simultaneously estimates the unmeasurable states and the uncertainties for the Electro-Hydraulic Actuator (EHA) system is presented. The main feature of the proposed observer is the use of an inter-sample output predictor which allows the users to increase the frequency acquisition of the piston position sensor without affecting the convergence performance. The stability analysis of the proposed observer is proved using Lyapunov function adapted to hybrid systems. To show the efficiency of our proposed observer, numerical simulations and experimental validation involving a control application which combines the designed observer and a PI controller for the purpose of piston position tracking problem is presented. Index Terms—Electro-Hydraulic Actuator (EHA), Continuousdiscrete time observers, Sampled data measurements, Intersample output predictor, Disturbance observer (DOB).
I. I NTRODUCTION
D
UE to a high power to weight ratio and their ability to generate high torques/forces outputs, EHA system are widely used in several industrial applications ([1]-[5]). Despite this advantage, the EHA systems suffer from some drawbacks due principally to their structure. Indeed, the EHA systems are subject to various uncertainties such as model parametric variations [6][7], highly nonlinear dynamic behavior [8], potential faults such as internal leakage [9] and hard damage affecting their functioning. In the last years the increasing demand of high precision control for EHA systems render the development of advance controls methods necessary to meet the actual requirements in terms of tracking performance. Despite their actual dominance, the traditional Proportional Integral Derivative (PID) controllers are not robust enough to counteract the effect of the uncertainties affecting the EHA systems. Therefore, the focus of the researchers has been shifted towards developing nonlinear closed loop control methods in order to improve the tracking performance for Manuscript received March 31, 2015; revised August 5, 2015; accepted January 24, 2016. Copyright (c) 2016 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to
[email protected]. This work was supported by the CEREEV (Combustion Engine for RangeExtended Electric Vehicle) project which is funded by the European Unions INTERREG IVA France-Manche-England programme. Dr Sofiane Ahmed Ali (
[email protected]) (Corresponding author) M Arnaud Christen (
[email protected]) and HDR Nicolas Langlois (
[email protected]) are with the IRSEEM/ESIGELEC Technople du Madrillet 76801 Saint Etienne du Rouvray, France. Dr Steven Begg (
[email protected]) is with the University of Brighton Cockcroft Building M24 Lewes Road BRIGHTON BN2 4GJ
the EHA systems. In the past decades, several nonlinear control technique have been developed in the literature such as feedback linearization [7][10], sliding mode control ([11][14]). In [6], a novel integration of adaptive control and integral robust feedback was proposed for hydraulic systems with considering all possible modeling uncertainties, and an excellent tracking performance was achieved, which is the first solution for theoretically asymptotic stability with unmatched disturbances for hydraulic systems, others nonlinear controllers such as robust/adaptive robust controllers ([15][20],[37][38]) and backstepping control ([21]-[24]) were also proposed. These methods have already proved their efficiency to improve the tracking performance of the EHA systems facing modeling uncertainties, parametric variations and external disturbances. However, all aforementioned techniques are full state feedback ones, i.e the designed controllers assume that all states of the EHA systems are available for measurements. From practical of point of view, this assumption may not be realistic for some hydraulic systems. Indeed, for many hydraulics applications only the position signal of the actuator is measured via sensor. The other states like velocity and hydraulic pressure are not measured because of the cost-reduction and the space limitation, therefore states and disturbances observers have received recently in the literature more and more attention. Several states and disturbances observers were developed by some researchers in the past decade. The idea behind developing these observers is to use the states and the disturbances estimation provided by these observers in order to synthesized an output- feedback controllers which compensate the internal and the external disturbances affecting the EHA systems. At this stage we can distinguish between two main approaches in the literature. The first approach consists in developing only a state estimator (i.e an observer) which estimates the unmeasurable state of the EHA systems. These observers ignore both the internal disturbances like parametric variations, modeling uncertainties, and the external disturbances such as the load and the friction torque affecting the hydraulic application. Those types of observers can be found in the work developed by the authors in ([25]-[28]). The second approach developed by the authors in ([29]-[31]) assumes that the states of the EHA systems are measurable and synthesize a Disturbance Observer (DOB) which estimates the mechanical and the hydraulic disturbances affecting the system. These estimations are incorporated then in a nonlinear closed loop controller which compensates the effect of the disturbance and
Copyright (c) 2016 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing
[email protected].
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TIE.2016.2531022 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS
improves the tracking performance of the desired position for the EHA systems. Recently, the authors in [32] proposed a novel framework for the purpose of simultaneous estimation of the unmeasurable states and the unmodeled disturbances, and then resulting in an excellent output feedback nonlinear robust backstepping controller for hydraulic systems, by developing an Extended State Observer (ESO) [33] and robust backstepping design. In this work, the authors consider that the main uncertainties affecting the EHA systems come from the hydraulic part. Therefore, the synthesized an observer based upon the wellknown techniques of extended state observers [33] which estimates the unmeasurable state and the hydraulic disturbances of the EHA systems. The proposed observer is also robust facing the mechanical disturbances generated by the load driven by the considered EHA system in the paper. In the case of hydraulic applications, the main drawback of the designed observers [25]-[32] is that they assume that the measured variable is continuous. In practical situations this measured variable which is given by the position sensor is sampled. In other words, the piston positions are available for the observer at only sampling times tk fixed by the sampling rate (i.e the frequency acquisition) of the sensor. This frequency can affects the convergence of the proposed when it comes to the matter of implementation of the proposed observer on Digital Signal Processors (DSP). Following the design in [32], the authors in ([34]) designed a sampled data observer which deals with the problem of discrete time-measurements for the EHA system. The proposed observer retains the same benefits which characterize the observer proposed in ([32]) in terms of simultaneous estimation of the unmeasurable states and the internal disturbances affecting the EHA system. The proposed observer involves in its structure an inter-sampled output predictor ([35]) which ensures continuous time estimation of the states and the exponential convergence of the observation errors. Moreover, the sampling period of the data acquisition of the observer, can be augmented independently from the frequency acquisition of the sensor position without affecting the convergence of the observer. However, the designed observer in ([34]) suffers from two major drawback. The first one concerns the Lyapunov function provided to prove the exponential convergence of the proposed observer. Indeed, the authors in ([34]) demonstrated the exponential convergence of the observer only locally between two sampling periods. In addition, the performance of the proposed observer were validated only in simulations and no experimental validation of the observer is provided. Comparing to the work of the author in ([34]), two main contributions were provided. The first contribution consists in designing a novel Lyapunov function based on small gain arguments which guaranty a global exponential convergence of the proposed observer. In addition the maximum sampling period Tmax derived from this function is less restrictive comparing to the one derived in ([34]). The second one is that experimental results performed on the experimental test rig of the Brighton university is provided for this observer. This is in our acknowledged the first time that such observer were designed and tested experimentally for the EHA systems.
2
The rest of the paper is organized as follows: the EHA modeling issues and the problem formulation are presented in Section II. Section III presents the continuous-discrete time observer for the EHA system. Numerical simulations and experimental validation showing the effectiveness of our proposed observer are presented in Section IV. Section V contains the conclusion and the future works. II. EHA MODELING The schematic of the electro-hydraulic actuator studied in this paper is depicted in Fig.1 [26][29]. The EHA system contains usually three parts namely the electrical, the mechanical and the hydraulic part. These parts represent an interconnected subsystem in such a way that the dynamic of each subsystem influences the dynamics of the others. The electrical part of the EHA system is a servo-valve (top of Figure 1) which controls the fluid dynamics inside the chambers. The spool valve is driven by the electrical input current u of a torque motor. The displacement of the spool valve xv together with the load pressure PL controls the fluid dynamic inside two chambers A and B which constitute the hydraulic part of the EHA system. The mechanical part of the EHA system is a cylindrical piston which is modeled as a classical mass-spring system. The position of the cylindrical piston xp , obeys to the fundamental principle of dynamics.
Fig. 1: The schematic of the EHA
A. State-space representation of the EHA Considering the following states variable: x = [x1 , x2 , x3 ]T = [xp , x˙p , PL ]T , the state-space representation of the EHA system can be written under the following form ([26],[29],[31]): x˙ 1 = x2 k b Ap (1) x˙2 = − x1 − x2 + x3 m m pm x˙3 = −αx2 − βx3 + γ Ps − sign(u)x3 u Where xp is the piston position [m].x˙ p [m/s] is the piston velocity and PL [P a] is the pressure load inside the chambers
Copyright (c) 2016 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing
[email protected].
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TIE.2016.2531022 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS
3
of the hydraulic part. k is the load spring constant [N/m] b is the viscous damping coefficient [N/(m/s)] and Ap is the cylinder bore m2 . Ps is the supply pressure [P a].α, β, γ are the hydraulic coefficients of the EHA model. These coefficients depend on the flow characteristics of the EHA system. For more details about the expression of the hydraulic coefficients α, β, γ and the modeling issues of the EHA system, the reader is referred to the work of the authors ([26],[29]) and their corresponding literature. B. Modeling uncertainties and time-varying disturbances affecting the EHA system In ([29],[31]), the authors distinguished between two types of disturbances d1 and d2 which can affect the EHA system. The first one d1 is the mechanical disturbance which is the result of lumping together the modeling parametric uncertainties, the load charge FLoad and the friction force Ff riction acting on the mechanical part of the EHA system. As reported by the authors in ([32]), the second term d2 does not hold the same significance as d1 . Indeed, d2 represents the parametric deviation over the hydraulic coefficients α, β, γ and potential leakage affecting the hydraulic device of the EHA system. These parameters are also sensitive to temperature inside the EHA system. Taking into account these issues, the disturbed EHA model can be written as follows ([29]): x˙ 1 = x2 b Ap d1 k x3 − x˙2 = − x1 − x2 + m m m m p x˙3 = −αx2 − βx3 + γ Ps − sign(u)x3 u + d2
(2)
Where d1 (t) and d2 (t) are expressed as follows ([31]) : d1 (t)
=
d2 (t)
=
k b Ap x1 − 4 x2 − 4 x3 + FLoad + FF riction (3) m m pm −4αx2 − 4βx3 + 4γ Ps − sign(u)x3 u −4
where x ∈ R3 and y ∈ R represent respectively the state vector and the measured piston position x1 = xp . The vector u ∈ R describes the set of admissible inputs. d(t) ∈ R2 denotes the vector of the disturbances which affect the EHA. Bd with dimensions 3 × 2. The matrices A,Bd ,C and vector φ(x, u) have the following structure: 0 1 0 A = 0 0 Amp 0 0 0 0 0 0 Bd = −1 m 0 1
0 b k x1p− m x2 −m ϕ(x, u) = −αx2 − βx3 + γ Ps − sign(u)x3 u
C. Problem formulation For system (4), the piston position is available for measurement only at each sampling times tk imposed by the frequency acquisition (the sampling period) of the sensor manufacturer. In this paper, we have to design a robust sampled data observer which simultaneously estimates the unmeasurable states x2 , x3 and the hydraulic disturbance term d2 of system (4). The designed observer must deals with the sampling phenomenon of the measured piston position xp and must be robust facing the mechanical disturbance term d1 (t). Under these considerations system (4) is rewritten as follows:
x˙ = Ax + ϕ(x, u) + Bd d y(tk ) = Cx(tk ) = x1 (tk )
(5)
System (5) combines a continuous dynamic behavior for the states x1 , x2 , x3 between two sampling times [tk , tk+1 ] and an updated step for the state x1 which occurs at the sampling times t = tk . III. C ONTINUOUS - DISCRETE TIME OBSERVER DESIGN FOR THE EHA SYSTEM In this section, we design a continuous-discrete time observer for the EHA system. Since d2 is the main disturbance term, we use the well-known technique of the augmented state system in order to estimate it. Following this, we add an extended variable x4 = d2 such as x˙ 4 = h(t) to system (5) so that the augmented state system can be written as follows:
The 4 symbolize the considered parametric uncertainties affecting the mechanical and the hydraulic part of the EHA system. System (2) can be expressed under the following compact form: x˙ = A x + ϕ(x, u) + Bd d (4) y = C x = x1
1 0 0
C=
¯x + ϕ(¯ x, u) + δ(t) x ¯˙ = A¯ y = C¯ x ¯ = x1
where x ¯ = [x1 , x2 , x3 , x4 ] and
0 0 A¯ = 0 0
1 0 0 0
0 Ap m
0 0
0 0 1 0
0 b k −m x1p− m x2 ϕ(¯ x, u) = −αx2 − βx3 + γ Ps − sign(u)x3 u 0 δ(t) =
C¯ =
0 −d1 m
0 h
1 0 0 0
Copyright (c) 2016 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing
[email protected].
(6)
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TIE.2016.2531022 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS
4
A. Observer design In this paper, our proposed observer will be designed under the same assumptions taken in [32]: Assumption 1. The disturbance term d1 (t) is bounded by a real unknown constant µ1 such that (|d1 (t)| < µ1 ) and the function h(t) is bounded by a real unknown constant µ2 such that (|h(t)| < µ2 . Remark 1. This assumption means that the mechanical disturbance and the derivative of the hydraulic disturbances affecting the EHA system are bounded by some unknown constants. From a practical point of view, the EHA system is a physical system which is BIBS (bounded input bounded state). So it is quite reasonable to consider such assumption. Assumption 2. In their practical range of parametric variak b x, u) = − m x1 − m x2 and ϕ3 (¯ x, u) = tions, the functions pϕ2 (¯ −αx2 − βx3 + γ Ps − sign(u)x3 u are locally (inside compact set) Lipschitz with respect to (x1 , x2 , x3 ) i.e. ∃β0 > 0, such that: |ϕ(X, u) − ϕ(Y, u)| ≤ β0 kX − Y k,
(i = 2, 3).
(7)
exists two matrices P, Q such that the following Lyapunov function is satisfied: ¯ + (A¯ − K C) ¯ T P ≤ −µIn P (A¯ − K C) where µ > 0 is a free positive constant and P is a symmetric positive definite matrix. Remark 3. Comparing to the work of the authors in ([26],[32]), the novelty in the designed observer (8) is the introduction of the inter-sample output predictor term w(t) ([35]) in the correction term. The dynamic of this predictor is simply a copy of the dynamics of system states equations. The role of the output predictor term is to provide a continuous time prediction of the output measured variable y(t). Indeed, since the measured output variable y(t) is sampled, its values y(tk ) are available for the observer only at sampling times t = tk . Comparing to constant-gain zero-order-hold (ZOH) approaches which maintain y(tk ) constant between the sampling times, the output predictor term w(t) will provide a continuous time estimation of y(t) as it is the case in continuous time observer design framework. Now we are able to state the main results of this paper:
Remark 2. At this point we mention that the function ϕ2 (¯ x, u) is globally Lipschitz withe respect to x2 , x3 . The function ϕ3 (¯ x, u), is differentiable everywhere except at u = 0, however and as stated by the authors in [32], this function is continuous and its derivative exists in the left and the right side of u = 0 and it is finite. Hence we can find a compact set so that ϕ3 (¯ x, u) is locally Lipschitz.
Theorem 1. Consider the EHA system (6), and suppose that assumptions (1-2) holds, given a sampling period T , √1 (θ+β0 ) choose σ0 , σ1 , σ2 as in (17), define σ3 = T eσT 2σ
Based on ([35]), let us consider the following continuousdiscrete time observer:
real positive bounded Tmax satisfying inequality (34), so that for all T ∈ (0, Tmax ), the radius of the ball can be made as small as desired by choosing large values of θ and ki=1,..,4 .
σ0
λmin (P )
then system (8) is an exponential sampled data observer for system (6) with the following properties: the vector of the observation error k¯ ex¯ k converges exponentially toward a ball whose radius R = √ 2σ2 . Moreover, there exists a σ0
λmin (P )(1−σ3 )
−1 ¯ˆ ˆ¯˙ ˆ¯ + ϕ(f (x ˆ = A¯x ¯), u) − θ4 ¯ − w(t)) x Proof 1. The proof of this theorem 1 is inspired from the θ K(C x ¯ ¯ ˆ ˆ ¯ + ϕ(f (x w(t) ˙ = C Ax ¯), u) t ∈ [tk , tk+1 ) k ∈ Nwork of the authors in ([35]). Let us now define the following observer ex¯ and the output ew (t) errors as following: w(tk ) = y(tk )) = x1 (tk ) (8) ˆ¯ − x ex¯ (t) = x ¯ (9) The function f is a saturation function which is introduced to ew (t) = w(t) − y(t) = w(t) − C¯ x ¯ ˆ guaranty that the estimated states x ¯ remains inside the compact Combining (6) and (8), we can easily check that for the EHA set so that the Lipschitz constant β0 always exists. The 4θ is ¯ system (6) the following properties is satisfied: θ4−1 θ A4θ = a diagonal matrix 4 × 4 defined by : −1 −1 ¯ ¯ ¯ θA and 4θ K C = 4θ K C4θ . Introducing the well-known 1 0 0 0 change of coordinate in the high gain literature e¯x¯ = 4θ ex¯ 0 1 0 0 yields to the following dynamics of the state and the output θ 4θ = 0 0 12 0 errors: θ 0 0 0 θ13 ¯ − K C¯ e¯x¯ + 4θ ϕ(f (x ˆ ˙ e ¯ = θ A ¯ ), u) − ϕ(¯ x , u) + x ¯ and the vector gains K ∈ R4×1 are chosen so that the matrix θKew − 4θ δ(t) ¯ is Hurwitz. The vector x ˆ (A¯ − K C) ¯ is the continuous-time e˙ w = θ¯ ex¯2 + ϕ1 (f (x ¯ˆ)), u) − ϕ1 (¯ x, u) estimate of the system state x ¯. The vector w(t) represents (10) the prediction of the output between two sampling times. The let us now consider the following candidate Lyapunov prediction w(t) is updated (re-initialised) at each sampling quadratic function V = e¯Tx¯ P e¯x¯ instant t = tk . 2 T T ˙
ˆ V ≤ −µθk¯ ex¯ k + 2¯ ex¯ P 4θ ϕ(f (x ¯), u) − ϕ(¯ x, u) + 2θ¯ ex¯ P Kew (t) −2¯ eTx¯ P 4θ δ (11)
B. Observability analysis ¯ C¯ in system (6), it can be From the structure of matrices A, ¯ C) ¯ is observable. Hence their easily checked that the pair (A,
Taking into account Assumptions (1-2) we have: V˙ ≤ −µθk¯ ex¯ k2 + 4β0 λmax (P )k¯ ex¯ k2 + 2θkP Kkk¯ ex¯ k|ew (t)| +4λmax (P )k¯ ex¯ kξ (12)
Copyright (c) 2016 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing
[email protected].
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TIE.2016.2531022 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS
5
p Where ξ = µ21 + µ22 Using the well-known property :
Which leads to:
2
λmin (P ) k¯ ex¯ k ≤ V ≤ λmax (P ) k¯ ex¯ k
2
eσt W (t) (13)
We derive: 4β0 λmax (P )V + + 2θkP Kk λmax (P ) λmin (P )
s
≤
M (t0 )
V |ew (t)| Now taking 0 < σ < σ0 /2 we derive: λmin (P ) s V ξ (14) +4λmax (P ) supt0 ≤s≤t (eσs W (s)) ≤ M (t0 ) λmin (P ) σ1 +2 supt0 ≤s≤t (eσs |ew (s)|) σ0 Now choosing the parameter θ such that θ > θ0 with θ0 = σ2 8β0 λ2 max (P ) sup{1, µλ } We have: +2 supt0 ≤s≤t (eσs ||ξ(s)||) min (P ) σ0 s and V V ˙ V ≤ −µθ + 2θkP Kk |ew (t)| σ1 2λmax (P ) λmin (P ) W (t) ≤ e−σt M (t0 ) + 2 supt0 ≤s≤t (e−σ(t−s) |ew (s)|) s σ0 V σ2 +4λmax (P ) ξ (15) +2 supt0 ≤s≤t (e−σ(t−s) ||ξ(s)||) λmin (P ) σ0 √ which leads to : consider now the function W = V then we got: M (t0 ) ||¯ ex¯ || ≤ e−σt p |ew (t)| W ˙ ≤ −µθ + θkP Kk p W λmin (P ) 4λmax (P ) λmin (P ) 2σ1 + p supt0 ≤s≤t (e−σ(t−s) |ew (s|) λmax (P ) +2 p ξ (16) σ0 λmin (P ) λmin (P ) 2σ2 supt0 ≤s≤t (e−σ(t−s) ||ξ(s)||) + p Let us set: σ λ 0 min (P ) µθ σ0 = 4λmax (P ) and θkP Kk (17) M (t0 ) σ1 = √ λmin (P ) supt0 ≤s≤t (eσs ||¯ ex¯ ||) ≤ p λmax (P ) λ min (P ) σ2 = 2 √ λmin (P ) 2σ1 σs + p supt0 ≤s≤t (e |ew (s)|) σ0 λmin (P ) Integrating (16), then 2σ2 Z t + p supt0 ≤s≤t (eσ(s) ||ξ(s)||) −σ0 (t−t0 ) −σ0 t σ0 s σ0 λmin (P ) W (t) ≤ e W (t0 ) + σ1 e e |ew (s)|ds (18) V˙ ≤ −µθ
V
(23)
σ1 supt0 ≤s≤t (eσs |ew (s)|) + σ0 − σ σ2 + supt0 ≤s≤t (eσs ||ξ(s)||) σ0 − σ
(24)
(25)
(26)
(27)
t
+σ2 e
−σ0 t
Z 0t
eσ0 s kξ(s)kds
t0
Multiplying both sides of (18) by eσt and using the fact that e−(σ0 −σ)t < 1 we derive: Z t eσt W (t) ≤ M (t0 ) + σ1 e−(σ0 −σ)t eσ0 s |ew (s|ds (19) +σ2 e
Z
−(σ0 −σ)t
t0 t
eσ0 s ||ξ(s)||ds t0
t
Z
Z
e(σ0 −σ)s eσs |ew (s|ds
eσt |ew (t)| ≤ eσt (θ + β0 ) (20)
t0 t
t
Z
tk
taking into account that e
e(σ0 −σ)s eσs ||ξ(s)||ds
−σs
e−σs ds suptk ≤s≤t (eσs k¯ ex¯ (s)k)ds (30)
< 1, we derive that:
suptk ≤s≤t eσs |ew (s)| ≤ T eσT (θ + β0 )suptk ≤s≤t (eσs k¯ ex¯ (s)k)ds (31)
t0
or eσt W (t) ≤ M (t0 ) Z
Multiplying again both sides of (28) by eσt and taking into account assumptions 1-2 we have: Z t eσt |ew (t)| ≤ eσt (θ + β0 ) e−σs eσs k¯ ex¯ (s)kds (29) Which leads to:
eσt W (t) ≤ M (t0 ) + σ1 e−(σ0 −σ)t
(21)
since suptk ≤s≤t (eσs k¯ ex¯ (s)) ≤ supt0 ≤s≤t (eσs k¯ ex¯ (s)) and taking into account that t > t0 , t1 , ...tk we derive that:
t
e(σ0 −σ)s ds supt0 ≤s≤t (eσs ||ew (s)||) t Z0 t −(σ0 −σ)t +σ2 e e(σ0 −σ)s ds supt0 ≤s≤t (eσs ||ξ(s)||)
+σ1 e−(σ0 −σ)t
tk
tk
where M (t0 ) = eσ0 t0 W (t0 ). In the other hand we have:
+σ2 e−(σ0 −σ)t
In the other hand we have from (10) the following expression of |ew (t)|: Z t ˆ |ew (t)| = |θ¯ ex¯2 + ϕ1 (f (x ¯), u) − ϕ1 (¯ x, u) |ds (28)
t0
supt0 ≤s≤t eσs |ew (s)| ≤ T eσT (θ + β0 )supt0 ≤s≤t (eσs k¯ ex¯ (s)k)ds (32) (22)
Combining (32) with (27) we have:
Copyright (c) 2016 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing
[email protected].
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TIE.2016.2531022 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS
6
TABLE I: NUMERICAL PARAMETERS VALUES FOR THE EHA SYSTEM
M (t0 ) supt0 ≤s≤t (eσs ||¯ ex¯ ||) ≤ p λmin (P ) +T eσT
(33)
parameters m b k Ap kv α β γ Ps
2σ1 p (θ + β0 )supt0 ≤s≤t (eσs k¯ ex¯ (s)k)ds) σ0 λmin (P ) 2σ2 + p supt0 ≤s≤t (eσ(s) ||ξ(s)||) σ0 λmin (P )
setting σ3 = T eσT
2σ1 (θ+β0 )
√
σ0
λmin (P )
then selecting Tmax satisfying the
following the small gain condition: Tmax eσTmax
2σ1 (θ + β0 ) p
